How can I create a rig where each bone only affects its surrounding mesh?
You have 2 solutions:
Parent With Empty Group instead of With Automatic Weight, then in the Properties panel &gt; Object Data &gt; Vertex Groups, assign the selected vertices to the vertex groups they are supposed to be part of (the groups that have the name of the bones that are supposed to control them).
Or select your object, go in Edit mode, select all, separate your object in several different objects wit P &gt; By Loose Parts, then back to Object mode, select a limb, shift select the armature, switch it to Pose mode, and parent to &gt; Bone.